ProGMAT wrote:
If x is a positive integer, is sqrt(x) an integer?
(1) sqrt(4x) is an integer.
(2) sqrt(3x) is not an integer.
Target question: Is sqrt(x) an integer?
Given: x is a positive integer
Statement 1: sqrt(4x) is an integer
IMPORTANT CONCEPT:
If K is an integer, then sqrt(K) will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
sqrt(144) = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
sqrt(1600) = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
sqrt(441) = 21 (integer), and 441 = (3)(3)(7)(7)[two 3's and two 7's]
sqrt(12) = some non-integer, and 12 = (2)(2)(3)[two 2's and
one 3's]
So, if sqrt(4x) is an integer, then the prime factorization of 4x has an even number of each prime.
Since 4x = (2)(2)(x) we can see that the prime factorization of x must have an even number of each prime.
If the prime factorization of x has an even number of each prime, then
sqrt(x) must be an integer.
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: sqrt(3x) is not an integer.
There are several values of x that meet this condition. Here are two:
Case a: x = 4. This means that sqrt(3x) = sqrt(12), which is not an integer. In this case,
sqrt(x) is an integer.
Case b: x = 5. This means that sqrt(3x) = sqrt(15), which is not an integer. In this case,
sqrt(x) is not an integer.
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer =
A
Cheers,
Brent
